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Why do we use 10 as a whole unit when it can't be evenly divided by 3 or 6 without resulting in an infinite decimal I.e 3.3333etc.

If units of 12 were used I.e 1 2 3 4 5 6 7 8 9 ? # 10. 3.3333 would become 3.4 (10/3=3.4)

In math, I'm told, if you have an infinite, something is wrong with the equation

So using a base of ten must be an incorrect way to measure

My apologies for the crudeness of this inquiry, I just typed it out on an iPod at a coffee shop real quick, little consideration for how articulate it comes out

David Thomas
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    Hexadactyly is rare. – Brian M. Scott Nov 15 '13 at 22:29
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    The Mayans used $60.$ look what happened to them. – Will Jagy Nov 15 '13 at 22:29
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    I am offended at the notion that there is something wrong with infinity :-) – Trevor Wilson Nov 15 '13 at 22:30
  • The Sumerians and Babylonians used sexigecimal (base 60), and for this reason, hours have 60 minutes, minutes have 60 seconds, and circles have 360 degrees. – Sammy Black Nov 15 '13 at 22:31
  • for any base you choose, some fractions will have infinite decimal expansions – ant Nov 15 '13 at 22:32
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    As far as I can tell, the Mayans used a strange almost place-value system based on the number $20$. I write "almost" because they represented numbers as $a_0 + a_1 \cdot 18 + a_2 \cdot 20^2 + a_3 \cdot 20^3 + \cdots$. The surprising $18$ was to make the decimal number $360$ (a good approximation of the number of days in a year) look particularly nice in their number system: $100$. – Sammy Black Nov 15 '13 at 22:34
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    @Will: No, the Maya used $20$ (except in calendrical computation, for which they modified it to make the equivalent of $100$ equal to $360$ instead of $400$. You are perhaps thinking of the Babylonians. – Brian M. Scott Nov 15 '13 at 22:36
  • Why don't we all just use continued fractions? I bet if everyone had to perform calculations on continued fractions to buy groceries the world would have been a whole lot smarter. – Alfonso Fernandez Nov 15 '13 at 22:39
  • You wouldn't solve the problem of $1/7$. When in math you find something infinite, you don't worry and go on. – egreg Nov 15 '13 at 22:59
  • @David Thomas I think Thomas Jefferson or Benjamin Franklin (I forgot which) might have advocated base 12. – Stefan Smith Nov 16 '13 at 01:29
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    @StefanSmith The Dozenal Society of Great Britain and The Dozenal Society of America have a fair number of members advocating for base 12 in the Anglosphere today. – Mark S. Dec 10 '13 at 19:08

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Most things that are meaningful in mathematics are completely independent of representation - it's important that the notation doesn't affect the result. The fact that we don't have $12$ units is just a convention, mostly based on the fact that we have $10$ fingers.

Besides, base $12$ doesn't allow division by $5$ without an infinitely repeating expansion, so it doesn't really address your objection that some numbers have expansions that don't terminate.

In fact, almost every real number has a non-terminating expansion regardless of whether you're using a system based on $10$ units or $12$.

  • I was having trouble dividing by 5 using 12 units of measurement in my head, so that does make sense, what would the repetitive decimal be if 12 were divided by 5 where 0.6 represents 1/2 of 1.0? – David Thomas Nov 16 '13 at 19:20
  • Also, if every number used results in a non- terminating expansion, doesn't this mean there is something fundamentally wrong with math itself, almost comparable to relativity vs quantum mechanics, only string theory and extra dimensions were concieved to balance that out a little. – David Thomas Nov 16 '13 at 19:26
  • There are many human civilizations that didn't use base 10. I don't think the people in those civilizations had extra or missing fingers to base their math system. – Neil Apr 11 '15 at 00:00